# Defining a topology on the plane

1. Define a new topology on the plane as follows. If p is a point on the plane, we say that U is a neighborhood of p if every straight line that contains p also contains an open line segment that is contained in U. Compare this topology with the usual topology on the plane (i.e., is it strictly weaker? strictly coarser? neither? are the two topologies equivalent?)

I am trying to define this topolgy.
I have $$\tau=\{U \subseteq X |\text{ if } (a,b)\cap p\in U, \text{ then } U \in N_{p}\}$$ where $$N_{p}$$ is the neighborhood system at p.
Is this correct? If so, am I writing it right? Something about it feels off.